Designing Efficient Parallel Algorithms for Graph Problems

Abstract

Graph algorithms are concerned with the algorithmic aspects of solving graph problems. The problems are motivated from and have application to diverse areas of computer science, engineering and other disciplines. Problems arising from these areas of application are good candidates for parallelization since they often have both intense computational needs and stringent response time requirements. Motivated by these concerns, this thesis investigates parallel algorithms for these kinds of graph problems that have at least one of the following properties: the problems involve some type of dynamic updates; the sparsi cation technique is applicable; or the problems are closely related to communications network issues. The models of parallel computation used in our studies are the Parallel Random Access Machine (PRAM) model and the practical interconnection network models such as meshes and hypercubes. Consider a communications network which can be represented by a graph G = (V;E), where V is a set of sites (processors), and E is a set of links which are used to connect the sites (processors). In some cases, we also assign weights and/or directions to the edges in E. Associated with this network, there are many problems such as (i) whether the network is k-edge (k-vertex) connected with xed k; (ii) whether there are k-edge (k-vertex) disjoint paths between u and v for a pair of given vertices u and v after the network is dynamically updated by adding and/or deleting an edge etc; (iii) whether the sites in the network can communicate with each other when some sites and links fail; (iv) identifying the rst k edges in the network whose deletion will result in the maximum increase in the routing cost in the resulting network for xed k; (v) how to augment the network at optimal cost with a given feasible set of weighted edges such that the augmented network is k-edge (k-vertex) connected; (vi) how to route messages through the network eAEciently. In this thesis we answer the problems mentioned above by presenting eAEcient parallel algorithms to solve them. As far as we know, most of the proposed algorithms are the rst ones in the parallel setting. Even though most of the problems concerned in this thesis are related to communications networks, we also study the classic edge-coloring problem. The outstanding diAEculty to solve this problem in parallel is that we do not yet know whether or not it is in NC. In this thesis we present an improved parallel algorithm for the problem which needs O( 4:5 log log n + 4 log n) time using O(n2 + n 3) processors, where n is the number of vertices and is the maximum vertex degree. Compared with a previously known result on the same model, we improved by an O( 1:5) factor in time. The non-trivial part is to reduce this problem to the edge-coloring update problem. We also generalize this problem to the approximate edge-coloring problem by giving a faster parallel algorithm for the latter case. Throughout the design and analysis of parallel graph algorithms, we also nd a